| L(s) = 1 | + 2.59i·3-s + 5-s + (−2.28 + 1.33i)7-s − 3.73·9-s − 3.05·11-s − 6.11·13-s + 2.59i·15-s − 1.25i·17-s + 3.37i·19-s + (−3.47 − 5.92i)21-s − 4.24i·23-s + 25-s − 1.92i·27-s − 1.52i·29-s + 9.46·31-s + ⋯ |
| L(s) = 1 | + 1.49i·3-s + 0.447·5-s + (−0.862 + 0.505i)7-s − 1.24·9-s − 0.922·11-s − 1.69·13-s + 0.670i·15-s − 0.304i·17-s + 0.774i·19-s + (−0.757 − 1.29i)21-s − 0.884i·23-s + 0.200·25-s − 0.369i·27-s − 0.283i·29-s + 1.70·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.265 + 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1197975883\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1197975883\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.28 - 1.33i)T \) |
| good | 3 | \( 1 - 2.59iT - 3T^{2} \) |
| 11 | \( 1 + 3.05T + 11T^{2} \) |
| 13 | \( 1 + 6.11T + 13T^{2} \) |
| 17 | \( 1 + 1.25iT - 17T^{2} \) |
| 19 | \( 1 - 3.37iT - 19T^{2} \) |
| 23 | \( 1 + 4.24iT - 23T^{2} \) |
| 29 | \( 1 + 1.52iT - 29T^{2} \) |
| 31 | \( 1 - 9.46T + 31T^{2} \) |
| 37 | \( 1 + 3.12iT - 37T^{2} \) |
| 41 | \( 1 + 3.19iT - 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 7.89T + 47T^{2} \) |
| 53 | \( 1 + 12.5iT - 53T^{2} \) |
| 59 | \( 1 - 14.5iT - 59T^{2} \) |
| 61 | \( 1 - 0.274T + 61T^{2} \) |
| 67 | \( 1 + 10.7T + 67T^{2} \) |
| 71 | \( 1 + 7.23iT - 71T^{2} \) |
| 73 | \( 1 - 0.742iT - 73T^{2} \) |
| 79 | \( 1 + 10.1iT - 79T^{2} \) |
| 83 | \( 1 - 6.15iT - 83T^{2} \) |
| 89 | \( 1 - 5.78iT - 89T^{2} \) |
| 97 | \( 1 + 14.3iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.189077050766146615698074083137, −8.311528861556538688848006423490, −7.36350784003638370918517159779, −6.32779177377589581971824260200, −5.49575804158604057135164770697, −4.89529440133350732833669665030, −4.11457746823386930780907299100, −2.92063476317989615259607140744, −2.45408928844541301916201827705, −0.04162916401278334152435394395,
1.19025065728412580605052331550, 2.48576657970343910186449650973, 2.90709738237215182595845475779, 4.49949835614911699476229775286, 5.41585331267782017579355705388, 6.33429033840804051107576655592, 6.89707033738949797314601975492, 7.58108911610531441274364933983, 8.070395910794058295179426987157, 9.276174596685007258762000411131
